How to blur specific borders of an object?
In Cycles, you can create a transparent material that blurs anything behind it like this :



The Light Path 'Is Camera Ray' is used to mix between a Transparent and a Refraction shader - this ensures that the material does not cast a shadow.

The Refraction shader has its Normal linked to the Incoming Geometry socket and this ensures that no actual refraction will take place due to incident rays always striking the surface along the Normal. This result in the Refraction shader behaving as a 'Transparent' shader when Roughness is zero but by increasing the Roughness (with IOR set to a suitable value) you can vary the amount of blurring that occurs.

You'll need to ensure the Render Samples is set sufficiently high for the blurring to be smooth.

With creative use of Generated coordinates to vary the Roughness you can even create a material with variable blurriness over its surface. In this example the blurring plane is placed over the edge of the checkered floor with variable blurring from left to right :